Chelsea were much the better side in the first-half, but mistakes cost the away side as United's quick counter-attacks punished Frank Lampard's side on Sunday afternoon.
Marcus Rashford was the double, one from the spot and one midway in the second-half to send United well on their way to all three points.
Anthony Martial was also on the scoresheet after he was found free at the back post and tapped into an empty net.
But Dan James stole the headlines. Ole Gunnar Solskjaer gave the Welshman game time towards the end and the midfielder scored on his Premier League debut, and dedicated his goal to his late father who passed away a few weeks ago.
Manchester United 4-0 Chelsea
The stage was set. Old Trafford - 'The Theatre of Dreams'. Ole Gunnar Solskjaer versus Frank Lampard in the opening game of both sides' Premier League 2019/20 campaign.
Mason Mount arriving at Old Trafford after being handed his Premier League debut by Frank Lampard.
Tammy Abraham hitting the woodwork in the early stages as Chelsea controlled the first 15 minutes at Old Trafford.
Kurt Zouma fouling Marcus Rashford inside the box after the hosts broke, and all of Chelsea's hard work had been washed away as the home side were awarded a penalty.
Marcus Rashford smashing the ball into the roof of the net to send Old Trafford into delirium, as the power was too much for Kepa Arrizabalaga was too much after guessing the right way.
Marcus Rashford [right] celebrating with his teammates Anthony Martial [left] and Jesse Lingard [centre] as Manchester United headed for their first victory of the season.
Aaron Wan-Bissaka competing with Mateo Kovacic as the Blues looked to fight back in Frank Lampard's first Premier League game in his short managerial career.
Anthony Martial dealing a huge blow to Chelsea's chance of getting any points after tapping home in front of the Stretford End to double the hosts' lead.
Anthony Martial marking his return to the no.9 shirt with a goal, and Marcus Rashford joined him in his celebrations.
Mason Mount enjoyed a fine debut as he found dangerous spaces, and will have felt he could've done better with openings he created and was given at Old Trafford.
Manchester United celebrating as a unit in a fine win for Ole Gunnar Solskjaer as a brutal and deadly counter-attacking performance was too much for Che.sea
Luck wasn't on Chelsea's side as they were left exasperated at the four goals leaked in a game where they started so dominantly, and the facial expressions painted the picture of their emotions.
Dan James came on in the final stages and found the net in his Premier League debut and enjoyed his emotional day remembering his dad who passed weeks before his move to Manchester United was completed.
Frank Lampard was left on the sidelines helpless as weeks of preparation was blown away as the Blues were punished for their lack of ruthlessness in defence and attack.
Despite the damning defeat, the travelling supporters stayed until the end to voice their support to Frank Lampard, who showed his appreciation to those who travelled.